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Chapter 5: Problem 96

Write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for \(a_{n}\) to find \(a_{7}\), the seventh term of the sequence. \(0.0004,-0.004,0.04,-0.4, \ldots\)

Short Answer

Expert verified
The formula for the nth term for the series is \(a_{n} = 5(-\frac{1}{5})^{n-1}\) and the seventh term of the sequence is -\frac{1}{15625}.

Step by step solution

01

Identify the Common Ratio

This is a geometric sequence, so there's a common ratio between each term. Divide the second term (-1) by the first term (5) to find the common ratio, \(r = -\frac{1}{5}\).
02

Formula for nth term

For a geometric sequence, the formula for the nth term is \(a_{n} = a_{1}r^{n-1}\) where \(a_{1}\) is the first term and \(r\) is the common ratio. Here \(a_{1}=5\) and \(r=-\frac{1}{5}\), so the formula becomes \(a_{n} = 5(-\frac{1}{5})^{n-1}\).
03

Find the seventh term

To find the seventh term \(a_{7}\), substitute \(n = 7\) into the formula \(a_{n} = 5(-\frac{1}{5})^{n-1}\). This gives \(a_{7} = 5(-\frac{1}{5})^{7-1} = -\frac{1}{15625}\).

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Most popular questions from this chapter

Write the first six terms of the geometric sequence with the first term, \(a_{1}\), and common ratio, \(r\). \(a_{1}=-\frac{1}{16}, r=-4\)

Find the indicated term for the geometric sequence with first term, \(a_{1}\), and common ratio, \(r\). Find \(a_{30}\), when \(a_{1}=8000, r=-\frac{1}{2}\).

Determine whether each sequence is arithmetic or geometric. Then find the next two terms. \(-7,-2,3,8, \ldots\)

Write the first six terms of the geometric sequence with the first term, \(a_{1}\), and common ratio, \(r\). \(a_{1}=-1000, r=0.1\)

Determine whether each sequence is arithmetic or geometric. Then find the next two terms. \(3, \frac{3}{2}, \frac{3}{4}, \frac{3}{8}, \ldots\)
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